maximizing)expected)u=lity)for)stochas=c)...

Jian Li Ins)tute of Interdisiplinary Informa)on Sciences

Tsinghua University Aug. 2012

Maximizing Expected U=lity for Stochas=c Combinatorial Op=miza=on Problems

Joint work with Amol Deshpande (UMD) �

TexPoint fonts used in EMF.

[email protected]

ISMP 2012. Berlin

Inadequacy of Expected Value

Stochas=c Op=miza=on Some part of the input are probabilis=c Most common objec=ve: Op=mizing the expected value

Inadequacy of expected value: Unable to capture risk-‐averse or risk-‐prone behaviors

Ac=on 1: $100 VS Ac=on 2: $200 w.p. 0.5; $0 w.p. 0.5 Risk-‐averse players prefer Ac=on 1 Risk-‐prone players prefer Ac=on 2 (e.g., a gambler spends $100 to play Double-‐or-‐Nothing)

Inadequacy of Expected Value�

Be aware of risk!

St. Petersburg Paradox�

Expected U=lity Maximiza=on Principle

Expected U)lity Maximiza)on Principle: the decision maker should choose the ac=on that maximizes the expected u)lity�

Remedy: Use a u=lity func=on �

Proved quite useful to explain some popular choices that seem to contradict the expected value criterion An axioma&za&on of the principle (known as von Neumann-‐Morgenstern expected u=lity theorem).

Problem Defini=on Determinis=c version:

A set of element {ei}, each associated with a weight wi

A solu=on S is a subset of elements (that sa=sfies some property) Goal: Find a solu=on S such that the total weight of the solu=on w(S)=ΣiєSwi is minimized

E.g. shortest path, minimal spanning tree, top-‐k query, matroid base

Problem Defini=on Determinis=c version:

A set of element {ei}, each associated with a weight wi

A solu=on S is a subset of elements (that sa=sfies some property) Goal: Find a solu=on S such that the total weight of the solu=on w(S)=ΣiєSwi is minimized

E.g. shortest path, minimal spanning tree, top-‐k query, matroid base

Stochas=c version: wis are independent posi=ve random variable μ(): R+→R+ is the u=lity func=on (assume limx →∞μ(x)=0) Goal: Find a solu=on S such that the expected u=lity E[μ(w(S))] is maximized

Our Results THM: If the following two condi=ons hold

(1) there is a pseudo-‐polynomial =me algorithm for the exact versionof determinis=c problem, and

(2) μ is bounded by a constant and sa=sfies Holder condi&on |μ(x)-‐ μ(y)|≤ C|x-‐y|α for constant C and α≥0.5,

then we can obtain in polynomial =me a solu=on S such that E[μ(w(S))]≥OPT-‐ε, for any fixed ε>0

Exact version: find a solu=on of weight exactly K Pseudo-‐polynomial =me: polynomial in K Problems sa=sfy condi=on (1): shortest path, minimum spanning tree, matching, knapsack. �

Our Results If μ is a threshold func&on, maximizing E[μ(w(S))] is equivalent to maximizing Pr[w(S)<1] minimizing overflow prob. [Kleinberg, Rabani, Tardos. STOC’97] [Goel, Indyk. FOCS’99]

chance-‐constrained stochas&c op&miza&on problem [Swamy. SODA’11]

1

10 �

μ(x)�

Our Results If μ is a threshold func&on, maximizing E[μ(w(S))] is equivalent to maximizing Pr[w(S)<1] minimizing overflow prob. [Kleinberg, Rabani, Tardos. STOC’97] [Goel, Indyk. FOCS’99]

chance-‐constrained stochas&c op&miza&on problem [Swamy. SODA’11]

However, our technique can not handle discon=nuous func=on directly

So, we consider a con=nuous version μ’�

1

1 1+δ�0 �

μ'(x)�1

10 �

μ(x)�

Our Results Other U=lity Func=ons

Exponential

Our Results Stochas)c shortest path : find an s-‐t path P such that Pr[w(P)<1] is maximized

Previous results Many heuris=cs Poly-‐=me approxima=on scheme (PTAS) if (1) all edge weights are normally distributed r.v.s (2) OPT>0.5[Nikolova, Kelner, Brand, Mitzenmacher. ESA’06] [Nikolova. APPROX’10]

Bicriterion PTAS (Pr[w(P)<1+δ]>(1-‐eps)OPT) for exponen=al distribu=ons [Nikolova, Kelner, Brand, Mitzenmacher. ESA’06]

Our result Bicriterion PTAS if OPT= Const

s� t�

Uncertain length�

Our Results Stochas)c knapsack: find a collec=on S of items such that Pr[w(S)<1]>γ and the total profit is maximized

Previous results log(1/(1-‐ γ))-‐approxima=on [Kleinberg, Rabani, Tardos. STOC’97] Bicriterion PTAS for exponen=al distribu=ons [Goel, Indyk. FOCS’99] PTAS for Bernouli distribu=ons if γ= Const [Goel, Indyk. FOCS’99] [Chekuri, Khanna. SODA’00]

Bicriterion PTAS if γ= Const [Bhalgat, Goel, Khanna. SODA’11] Our result

Bicriterion PTAS if γ= Const (with a bemer running =me than Bhalgat et al.) Stochas=c par=al-‐ordered knapsack problem with tree constraints�

Knapsack, capacity=1 �Each item has a deterministic profit and a (uncertain) size �

Our Algorithm Observa=on: We first note that the exponen=al u=lity func=ons are tractable


Approximate the expected u=lity by a short linear sum of exponen=al u=lity Show that the error can be bounded by є


Approximate the expected u=lity by a short linear sum of exponen=al u=lity Show that the error can be bounded by є

By linearity of expecta=on

Our Algorithm Problem: Find a solu=on S minimizing


Suffices to find a set S such that Opt


Suffices to find a set S such that

(Approximately) Solve the mul=-‐objec=ve op=miza=on problem with objec=ves for k=1,…,L

Opt

Approxima=ng U=lity Func=ons

How to approximate μ() by a short sum of exponen=als? (with lim μ(x)-‐>0 )


How to approximate μ() by a short sum of exponen=als? (with lim μ(x)-‐>0 ) A scheme based on Fourier Series decomposi=on:

For a periodic func=on f(x) with period 2¼

where the Fourier coefficient


How to approximate μ() by a short sum of exponen=als? (with lim μ(x)-‐>0 ) A scheme based on Fourier Series decomposi=on:

For a periodic func=on f(x) with period 2¼

where the Fourier coefficient Consider the par=al sum (L=2N+1)

Approxima=ng The Threshold Func=on

y=µ(x)

Par)al Fourier Series Periodic !

Gibbs Phenomenon�

Need L=poly(1/²) terms �


y=µ(x)


Damping Factor : Biased approxima=on


y=µ(x)



Ini)al Scaling : Unbiased

Apply Fourier series expansion on �


y=µ(x)



Ini)al Scaling : Unbiased

Extending the func. Unbiased and bemer quality around origin

The Mul=-‐objec=ve Op=miza=on Want to find a set S such that

(Approximately) Solve the mul=-‐objec=ve op=miza=on problem with objec=ves for i=1,…,L

(Similar to [Papadimitriou, Yannakakis FOCS’00]) Each element e has a mul=-‐dimensional weight

The Mul=-‐objec=ve Op=miza=on Discre=ze the vector

Enumerate all possible 2L-‐dimensional vector A (poly many) Think A as the approximate version of

Check if there is any feasible solu&on S such that

We use the pseudo-‐poly algorithm for the exact problem

Summary We need L=poly(1/²) terms We solve an O(L) dim op=miza=on problem The overall running =me is

This improves the running =me in [Bhalgat, Goel, Khanna. SODA’11] �

Extension: Mul=-‐dimensional Weights Stochas)c Mul)dimensional Knapsack: Each element e is associated with a random vector (entries can be correlated) d =O(1) Each element e is associated with a profit pe Objec=ve: Find a set S of items such that

and the total profit is maximized

Our result: We can find a set S of items in poly =me such that the total profit is at least (1-‐²) of the op=mum and

Extension: Mul=-‐dimensional Weights Consider the 2-‐dimensional u=lity func=on

Consider the rectangular par=al sum of the 2d Fourier series expansion �

A con=nuous version of the 2d threshold func=on �

Recent Progress Other problems in P? min-cut, matroid base

There is no pseudo-poly time algorithm for EXACT-min-cut Why? EXACT-min-cut is strongly NP-hard (harder than MAX-CUT)

It is a long standing open problem whether there is a pseudo-poly-time algo for EXACT-matroid base

Can we get the same result for these problems? Not for min-cut Consider a deterministic instance where MAX-CUT=1

μ�

1 0.878

Recent Progress (1) Monotone and Lipschitz nonincreasing u=lity func=on

(2) PTAS for the determinis=c mul=-‐dimensional version

then we can obtain S such that E[μ(w(S))]≥OPT-‐ε,

for any fixed ε>0.

Determinis=c mul=-‐dimensional version:

Each edge e has a weight (const dim) vector ve.

Given a target vector V, find a feasible set S s.t. v(S)<=V

A PTAS should either return a feasible set S such that v(S)<=(1+ε)V or claim that there is no feasible set S such that v(S)<=V

Recent Progress KNOWN:

Pseudo-‐poly for EXACT-‐A => PTAS for MulDim-‐A [Papadimitriou, Yannakakis FOCS’00]

PTAS for MulDim-‐MinCut [Armon, Zwick Algorithmica06]

PTAS for MulDim-‐Matroid Base [Ravi, Goemans SWAT96]

Pseudo-poly for EXACT

PTAS for MulDim

Shortest path, Matching, MST, Knapsack

Min-cut

Matroid base?

Open Problems OPEN: Can we extend the technique to the adap=ve problem (considered in [Dean,Geomans,Vondrak, FOCS’04 ]

[Bhalgat, Goel, Khanna. SODA’11])?

OPEN: Can we get a PTAS? (with a mul=plica=ve error) Even for op=mizing the overflow probability

Consider maximiza=on problems and increasing u=lity func=ons

Op=mizing or For and , see [Kleinberg, Rabani, Tardos. STOC’97], [Goel, Guha, Munagala PODS’06] �

Thanks [email protected]

maximizing)expected)u=lity)for)stochas=c)...

Documents